Order Beam Theory Research Articles

Vibration of carbon nanotube reinforced composite beams based on the first and third order beam theories

This paper investigates the linear free vibration of nanocomposite beams reinforced by single-walled carbon nanotubes (SWCNTs). Two types of CNT reinforced beams, namely uniformly distributed CNT reinforced (UD-CNT) beams and functionally graded CNT reinforced (FG-CNT) beams, are considered. It is assumed that the SWCNTs are aligned along the beam axial direction and the distribution of the SWCNTs may vary through the thickness of the beam. The virtual strain and kinetic energies of the FG-CNT composite beam are obtained using the classic variational method of Hamilton’s principle and then solved by the p-Ritz method. Vibration frequency parameters for the FG-CNT beams based on the first order and third order beam theories are presented and the effects of CNT filler volume fraction, distribution, beam span to depth ratio and end support conditions on the free vibration characteristics of the beams are discussed. Comparison studies for UD-CNT and FG-CNT beams based on the first order and the third order beam theories are also performed and the differences in vibration frequencies between these two theories are highlighted.

Finite element analysis for buckling of two-layer composite beams using Reddy’s higher order beam theory

In this paper, a two-layer partial composite columns model is built based on Reddy׳s higher order beam theory, and two novel displacement based finite elements for this and Timoshenko composite beams are respectively formulated by means of the principle of minimum potential energy. Subsequently, the buckling analyses of pinned–pinned and clamped–guided composite columns are performed using the proposed finite elements, and the results are compared with those obtained by plane stress model, Timoshenko and Newmark composite beams model respectively. The superior quality of Reddy composite columns model over Timoshenko composite columns model and the correctness of the proposed Timoshenko composite columns model are demonstrated by the numerical comparison. Finally, the parametric study explores effects of parameters including stiffness of shear connectors, span-to-depth ratios, Young׳s modulus ratios and sub-layer׳s depth on the buckling load. The discrepancies between the performance of higher order and Timoshenko composite columns have also been numerically investigated.

Nonlinear vibration analysis of Timoshenko nanobeams based on surface stress elasticity theory

Abstract In this article, the nonlinear free vibration behavior of Timoshenko nanobeams subject to different types of end conditions is investigated. The Gurtin–Murdoch continuum elasticity is incorporated into the Timoshenko beam theory in order to capture surface stress effects. The nonlinear governing equations and corresponding boundary conditions are derived using Hamilton's principle. A numerical approach is used to solve the problem in which the generalized differential quadrature method is applied to discretize the governing equations and boundary conditions. Then, a Galerkin-based method is numerically employed with the aim of reducing the set of partial differential governing equations into a set of time-dependent ordinary differential equations. Discretization on time domain is also done via periodic time differential operators that are defined on the basis of the derivatives of a periodic base function. The resulting nonlinear algebraic parameterized equations are finally solved by means of the pseudo arc-length continuation algorithm through treating the time period as a parameter. Numerical results are given to study the geometrical and surface properties on the nonlinear free vibration of nanobeams.

Vibration analysis of a beam with embedded shape memory alloy wires

In this study, analytical relations for evaluating the exact solution of natural frequency and mode shape of beams with embedded shape memory alloy (SMA) wires are presented. Beams are modeled according to Euler-Bernoulli, Timoshenko and third order beam (Reddy) theories. A relation is obtained for determining the effect of axial load generated by the recovery action of pre-strained SMA wires. By defining some dimensionless quantities, the effect of different mechanical properties on the frequencies and mode shapes of the system are carefully examined. The effect of axial load generated by SMA wires with buckling load and frequency jump is accurately studied.

Quality Requirements for the Design, Fabrication and Repair of Welded Vessels

The Eurocode standards provide common structural design rules. The design is based on selecting minimum weight members to satisfy given conditions. The standard applicable to the design of various tanks, vessels, containers and bottles for gas and liquid is: EN 1993-4-2; Eurocode 3; Design of steel structures; Part 4.2: Tanks. For the verification of certain mechanical characteristics of such vessels, the applied standard is EN 1993-1-6; Supplementary rules for the strength and stability of shell structures. Verification of the specific actions on structures, that imply dead and live loads, must be also applied for the tanks and vessels, according to EN 1991 and related standards. Construction of welded vessels must be performed with qualified welding procedures. WPS/PQR and welder approvals (EN 287 and EN ISO 15614) are required. The longitudinal and circumferential joints must be butt-welded, both with full penetration. Tolerances, heat treatment and marking are indicated. Some examples of own realized welded vessels are presented. Manufacturing inspection, by visual examination, must cover most aspects: bending, groove preparation, alignment before and after welding, welding process, penetration, tolerances for circumference, straightness and circularity, double-sided welds, heat treatment, small diameter welds around the nozzles, detecting of undercut, excessive convexity, distortions, etc. Defect acceptance criteria are presented, depending on the weld types and quality level, according to EN ISO 5817. Macrographic examination according to EN 1321 of a transverse section of the welds, tensile test according to EN ISO 4136 and radiography testing of welds in accordance with EN 1435 class B are required. For pressure vessels, according to the ISCIR Technical Prescriptions, active in Romania, C 4-2010 „Static metal pressure vessels”, respective C12-2010 „Technical requirements for metal tank, container and barrel recipients for compressed, liquefied and solved gases, under pressure”, tightness and hydrostatic tests are performed at the maximal design pressure for 60 min. For the stability verification, the ultimate limit state of local shell buckling should be applied, according to EN 1993-1-6, by the conventional method, stress design concept or global second order beam theory. The fatigue strength of the structural shell must be verified by the methods given in EN 1993-1-6. Weld repairs are allowed, with no limit on the number and extent of defects. The repairs by welding are carried out according to ASME Code, Sections IX and VIII. According to the ISCIR Technical Prescriptions, a technology for repair welding must be elaborated by the manufacturer and approved by ISCIR. The design, fabrication and repairs of vessels must be performed according to the mentioned standards, depending on the class of the vessel.

Free Vibration Analysis of Cracked Composite Beams Subjected to Coupled Bending-Torsion Loads Based on a First Order Beam Theory

In this paper free vibration analysis of cracked composite beams subjected to coupled bending-torsion loads are presented. The composite beam is assumed to have an open edge crack. A first order theory is applied to count for the effect of the shear deformations on natural frequencies as well as the effect of coupling in torsion and bending modes of vibration. Local flexibility matrix is used to obtain the additional boundary conditions of the beam in the crack area. After obtaining the governing equations and boundary conditions, GDQ method is applied to solve the obtained eigenvalue problem. Finally, some numerical results are given to show the efficacy of the method. In addition, to count for the effect of coupling on natural frequencies of the cracked beams, different fiber orientations are assumed and studied.

A unified higher order beam theory for buckling of a functionally graded microbeam embedded in elastic medium using modified couple stress theory

Based on the modified couple stress theory (MCST), a unified higher order beam theory which contains various beam theories as special cases is proposed for buckling of a functionally graded (FG) microbeam embedded in elastic Pasternak medium. This non-classical microbeam model incorporates the material length scale parameter which can capture the size effect. The non-classical beam model reduces to the classical beam model when the material length scale parameter is set to zero. The material properties of the FG microbeam are assumed to vary in the thickness direction and are estimated through the Mori–Tanaka homogenization technique and the classical rule of mixture. The governing equations and the related boundary conditions are derived using the principal of the minimum total potential energy. The Navier-type solution is developed for simply-supported boundary conditions, and explicit expressions related to each type of beam theory are proposed for the critical buckling load. Numerical results are presented to investigate the influences the material length scale parameter, aspect ratio, different estimation method of material properties, various material compositions, and the parameters of the elastic medium on the critical buckling load. Comparison study is also performed to verify the present formulation.

Bending and vibration of functionally graded microbeams using a new higher order beam theory and the modified couple stress theory

Static bending and free vibration of functionally graded (FG) microbeams are examined in this paper based on the modified couple stress theory (MCST) and various higher order beam theories (HOBTs). This non-classical microbeam model incorporates the material length scale parameter which can capture the size effect. The material properties of the FG microbeams are assumed to vary in the thickness direction and are estimated through the Mori–Tanaka homogenization technique. The governing equations and the related boundary conditions are derived using Hamilton’s principle. The Navier-type solution is developed for simply-supported boundary conditions. Numerical results are presented to investigate the influences the material length scale parameter, different material compositions, and shear deformation on the bending and free vibration behavior of FG microbeams. Some of the present results are compared with the previously published results to establish the validity of the present formulation. It is established that the present FG microbeams exhibit significant size-dependence when the thickness of the microbeam approaches to the material length scale parameter.

Modal analysis of the FGM beams with effect of axial force under longitudinal variable elastic Winkler foundation

A differential equation of the homogenized functionally graded material (FGM) beam deflection and its solution will be presented which has been used in free vibration analysis of the beams with polynomial continuous longitudinal and transversal variation of material properties. The FGM beams are considered to be resting on longitudinal variable (Winkler) elastic foundation. Homogenization of the varying material properties and calculation of other parameters have been done by the multilayer method and the direct integration method. The 2nd order beam theory has been used for establishing the kinematic and kinetic equations. Not only the shear force deformation effect and the effect of consistent mass distribution and mass moment of inertia have been taken into account, but also the effect of axial forces have been considered. Numerical experiments were performed to calculate the eigenfrequencies and corresponding eigenmodes of chosen FGM beams. The solution results are discussed and compared with those obtained using a very fine mesh of solid finite elements. The effects of material properties and elastic foundation variation, the layering fineness, the average shear force correction factor and the axial forces have been evaluated and discussed.

Measurement of adhesive shear properties by short beam shear test based on higher order beam theory

This paper refers to the measurement of the shear properties of adhesive bonding by a new beam theory using the short beam shear (SBS) test configuration. A novel higher-order sandwich beam theory has been developed to analyze the adhesive bonded beam that consists of two adhered laminates and a single layer of adhesive in between. The closed form analytical solution for the SBS test model of the adhesively bonded beam is obtained in terms of deflection and stress distribution. The present theory has been used for calculating the adhesive shear modulus from the structural compliance. The initiation of stiffness degradation for the short beam shear test model was used as the critical load value for deriving the adhesive shear strength. A finite element model is built for validating the present model, and to evaluate its suitability for measuring adhesive shear properties. The present theory shows better accuracy for measuring the shear modulus than existing theories for both thin and thick adhesive layers. The measured strength values are more accurate than those obtained from the single lap joint shear test model. This theory can be used for adhesive materials with linear elastic deformation behavior.

The effect of transverse normal strain in contact of an orthotropic beam pressed against a circular surface

The contact problem of a straight orthotropic beam pressed onto a rigid circular surface is considered using beam theories that account for transverse shear and transverse normal deformations. The circular nature of the rigid surface emphasizes the difference between Euler Bernoulli theory behavior, where point loads develop at the edge of contact, and the higher order theories that predict non-singular pressure distributions. While Timoshenko beam theory is the simplest theory that addresses this behavior, the prediction of a maximum value of pressure at the edge of contact contradicts the elasticity theory result that contact pressure must drop to zero. Transverse normal strain is therefore introduced, both to study this fundamental discrepancy and to include an important effect in many contact problems. To investigate this effect, higher order beam theories that account for both constant and linear transverse normal strain through the beam thickness are derived using the principle of virtual work. The resulting orthotropic beam theories depend on the bending stiffness (EI), shear stiffness (GA), axial stiffness (EA1) and transverse normal stiffness (EA2), which are independent stiffness parameters that can differ by orders of magnitude. The above mentioned contact problem is then solved analytically for these theories, along with the Timoshenko beam model which assumes zero transverse normal strain. The results for different orthotropic materials show that inclusion of transverse normal deformation has a significant effect on the contact pressure solution. Furthermore, the solution using higher order beam theories encompasses the two extremes of a Hertz-like contact pressure when the half contact length is smaller than the thickness of the beam, and the Timoshenko beam theory case when the half contact length is much larger than the thickness. Concerning the behavior of the pressure at the edge of contact, adherence to the boundary conditions required by the principle of virtual work, shows that while the pressure does tend to zero, it does not become zero unless artificially enforced. In this regard the solution for the case of linear strain is better than that for constant strain. All beam solutions are validated with plane elasticity solutions obtained using the commercial finite element software ABAQUS.

An Influence Analysis of Second-Order Effect to the Vibration Control of Piezoelectric Beam by the Spline Finite Point Method

The field functions of the spline finite point (SFP) method were constructed by the linear combination of B-spline basis function, and the higher-precision results would be obtained with less discrete nodes by the SFP method. In this paper, based on Reddy's third order beam theory, a motion equation was developed by the SFP method to analyze the first five natural frequencies of piezoelectric laminated beam under different axial forces. The influence of second-order effect caused by the axial force on vibration control was discussed based on the modal control theory and the Linear Quadratic Regulator (LQR) optimal control method. It can be concluded that the SFP method is suitable for the dynamic analysis of piezoelectric beam which needs less computational cost and has high accuracy. The vibration of the structure can be effectively inhibited by the LQR method and modal control theory. And the axial force has significant impact on the natural frequencies and control voltage of piezoelectric beam.

A new two-dimensional refined higher order theory for in-plane free vibrations of sandwich and composite thick curved beams with flexible cores

Closed-form formulations of two-dimensional refined higher order beam theories and higher order beam theories for the free vibration analysis of simply supported cross-ply laminated composite and sandwich curved beams are presented. The present refined higher order beam theory analysis incorporates a trapezoidal shape factor that arises due to the fact that the stresses through the thickness of the beam are integrated over a curved surface. Therefore, the present theory refines other higher order beam theories established hitherto for free vibration analysis of strongly thick circular curved beams. The stress–strain relationship is derived from an orthotropic lamina in a three-dimensional state of stress. Numerical results are presented for the natural frequencies of laminated composite and sandwich thin/thick straight beams and shallow/deep and moderately/strongly thick curved beams. The closed-form solutions presented herein are compared with the available exact two-dimensional elasticity and analytical solutions in the available literature and excellent agreement is obtained.

Analysis of composite beams with longitudinal and transverse partial interactions using higher order beam theory

A new one dimensional finite element model based on a higher order beam theory is presented for the analysis of composite beams taking into account the effect of longitudinal as well as vertical partial interaction between the adjacent layers. The proposed method models the transverse shear deformation of the beam components in a refined manner. A third order variation of the axial displacement of the fibres over the beam depth is taken to have a parabolic variation of shear stress which is also made zero at the beam top and bottom surfaces. In the proposed FE model, there is no need of incorporating any shear correction factor and the model is free from shear locking problem. In addition to correctly predicting the global responses of the beam, the model can predict better distribution of stresses than the existing models based on Euler–Bernoulli or Timoshenko beam theory. Many new results are presented as there is no published result on the present problem based on higher order beam theory.

Analysis of composite beams with partial shear interactions using a higher order beam theory

A new finite element model based on a higher order beam theory is presented for the analysis of composite beams. The proposed model takes into account the effect of partial shear interaction between the adjacent layers as well as transverse shear deformation of the beam. A third order variation of the axial displacement of the fibres over the beam depth is taken to have a parabolic variation of shear stress which is also made zero at the beam top and bottom surfaces. In the proposed FE model, there is no need of incorporating any shear correction factor and the model is free from shear locking problem. In addition to correctly predicting the global responses of the beam, the model can predict better distribution of stresses than the existing models based on Euler–Bernoulli or Timoshenko beam theory. The proposed finite element model is validated by comparing the results with those available in literature. Many new results are presented for future references as there is no published result on composite beams based on higher order beam theory.

Selection of the initial geometrical imperfection in nonlinear FE analysis of cold-formed steel rack columns

This paper presents three methodologies to predict the load carrying capacity of cold formed steel rack columns via nonlinear finite element analysis (FEA). The column lengths are chosen in the range where the main failure is due to distortional buckling. It is demonstrated that for this range, to derive an accurate prediction of the ultimate load is more complex than for lengths where the main failure is local or global. The nonlinear analysis is carried out introducing an initial geometric imperfection on the upright. The three methodologies presented here allow for different imperfection shapes. The first one uses the critical mode shape (the first buckling mode). The second corresponds to an iterative methodology in which the shape that leads to the lowest ultimate load is used. These two first methodologies use exclusively the finite element method (FEM). The third one combines the finite element analysis with the generalised beam theory (GBT) in order to determine the modal participation of the FEM buckling mode and generate a particular combined geometric imperfection. To validate the predicted loads, the results of the three methodologies are compared with values obtained in experimental tests.

Modal analysis of the FGM-beams with continuous transversal symmetric and longitudinal variation of material properties with effect of large axial force

In this contribution we deal with modelling and simulation of a free vibration of the 2D functionally graded material (FGM) beams with continuous spatial variation of material properties. The fourth-order differential equation of the second order beam theory has been presented which was used in modal analysis with effect of large axial force. The continuous variation of the effective elasticity modulus and mass density can be caused by continuous variation of both the volume fraction and material properties of the FGM constituents in the transversal and longitudinal direction. The polynomial variation of these parameters has been considered. Homogenization of the varying material properties and the calculation of other parameters have been done by two methods. In the first one, the extended mixture rules and laminate theory have been used, where the real composite beam has been transformed to the multilayer beam. In the second one, the direct integration method has been used. Not only the shear force deformation effect and the effect of consistent mass distribution and mass moment of inertia but also the effect of large axial force has been taken into account. Numerical experiments have been done concerning the calculation of the eigenfrequencies and eigenmodes of chosen FGM beams. Effect of division fineness of the beam to the layers in transversal direction on the solution results has been evaluated. The solution results obtained by the both methods have been discussed and compared with the ones obtained using very fine meshes of 2D solid elements of a FEM commercial program.

Assessment of nanotube structures under a moving nanoparticle using nonlocal beam theories

Dynamic analysis of nanotube structures under excitation of a moving nanoparticle is carried out using nonlocal continuum theory of Eringen. To this end, the nanotube structure is modeled by an equivalent continuum structure (ECS) according to the nonlocal Euler–Bernoulli, Timoshenko and higher order beam theories. The nondimensional equations of motion of the nonlocal beams acted upon by a moving nanoparticle are then established. Analytical solutions of the problem are presented for simply supported boundary conditions. The explicit expressions of the critical velocities of the nonlocal beams are derived. Furthermore, the capabilities of various nonlocal beam models in predicting the dynamic deflection of the ECS are examined through various numerical simulations. The role of the scale effect parameter, the slenderness ratio of the ECS and velocity of the moving nanoparticle on the time history of deflection as well as the dynamic amplitude factor of the nonlocal beams are scrutinized in some detail. The results show the importance of using nonlocal shear deformable beam theories, particularly for very stocky nanotube structures acted upon by a moving nanoparticle with low velocity.

Frequency response of pre-stressed structures with nonlinear cohesive interfaces

This paper explores two different methods to extract information about localized nonlinear shear cohesive behavior of interfaces from linear elastodynamic response. Both formulations are based on the concept of loading a cohesive interface statically into the nonlinear region and then superposing small amplitude vibrations or wave motions on the statically pre-loaded state. To a first order approximation the cohesive interface’s small amplitude dynamic behavior is assumed to be linearly compliant with the stiffness interpreted as the local slope of the cohesive law at the static pre-load level. The first investigation considers the vibrations of a beam made up of two sections partially bonded together through the depth of the beam to create a cracked interface. The dependence of the natural frequencies on the spring stiffnesses of a beam with the cracked interface replaced by bending and shear springs is calculated using first order beam theory. The dependence of the spring stiffnesses used in the vibration analysis on crack length, static pre-load, and yield stress of a Dugdale–Barenblatt cohesive zone in the interface is determined through a 2D elasto-static boundary element analysis of a cracked beam element, which treats the boundary nonlinearities directly. The second investigation concerns the dispersion relations for time-harmonic guided waves in a layer connected to a rigid substrate by a very thin interface layer of material with nonlinear and softening behavior. The interfacial spring stiffness which, is directly included in the dynamic layer boundary conditions, is again interpreted as the local slope of the cohesive law at the static pre-load level. The spring stiffnesses inferred from the dispersion relations of either SH or generalized Rayleigh–Lamb waves from a series of measurements taken at multiple pre-load levels could then be integrated to obtain the cohesive law.

An effective finite element for torsion of constant cross-sections including warping with secondary torsion moment deformation effect

In this paper an effective finite element of constant stiffness is developed for torsion with warping of both open- and closed-shape cross-sections. The secondary torsion moment deformation effect has been taken into account. The local element stiffness matrix that describes torsion with warping of both open- and closed-shape cross-sections has been derived using an analogy between torsion with warping (including the secondary torsion moment deformation effect) and the 2nd order beam theory (including the shear force deformation effect). The deformation effect of the secondary torsion moment must be taken into account first of all when dealing with closed-shape cross-sections. The warping part of the first derivative of the twist angle has been considered as an additional degree of freedom in each node at the element ends. This can be regarded as part of the twist angle curvature caused by the warping moment. Numerical results are presented to demonstrate the efficiency and accuracy of this new beam finite element. The necessity of including the deformation effect of the secondary torsion moment into the solution is demonstrated on the torsion of closed cross-sections. The results are compared with those obtained using other special commercial codes.